Moment polytopes on Sasaki manifolds and volume minimization

We show that transverse coupled Kähler–Einstein metrics on toric Sasaki manifolds arise as a critical point of a volume functional. As a preparation for the proof, we re-visit the transverse moment polytopes and contact moment polytopes under the change of Reeb vector fields. Then we apply it to a coupled version of the volume minimization by Martelli–Sparks–Yau. This is done assuming the Calabi–Yau condition of the Kähler cone, and the non-coupled case leads to a known existence result of a transverse Kähler–Einstein metric and a Sasaki–Einstein metric, but the coupled case requires an assumption related to Minkowski sum to obtain transverse coupled Kähler–Einstein metrics.

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Received 3 February 2022

Accepted 18 July 2022

Published 7 April 2023